Harmonic dipole oscillation

In the classical theory of scattering (Cohen-Tannoudji et al. 1977, James 1982), atoms are considered to scatter as dipole oscillators with definite natural frequencies. They undergo harmonic vibrations in the electromagnetic field, and emit radiation as a result of the oscillations. 

The effect can be described by classical mechanics in terms of forced vibrations of harmonic oscillators. Here since the molecular polarizability t. changes slightly as Ihe bond distorts, nonlinear effects give rise to dipole oscillations at frequencies other than the imposed frequency. Raman himself seems to have been led to this discovery, at least In part, by his theoretical studies of Ihe vibrations of musical instruments such as the violin. 

Dipole oscillations in an assembly of molecules in the membrane of cells can be modeled as phase-locked solid state oscillators by a basic circuit as in Figure 1. Loose coupling between such circuits imposes an eigenvalue problem from which significant mode softening can be shown to result and this has been suggested to be an important requirement in the energetics associated with the reproduction and mutation of cells. As each individual unit oscillator can operate at subharmonics as well as harmonics, the above model is consistent with the idea that in vivo a number of discrete frequencies exist in the cell. 

A harmonic oscillator is a particle oscillating along the x-axis under the action of a quasi-elastic force F = — x (where K is the elasticity coefficient and X is the displacement of the particle from its equilibrium position) with a potential energy given by V = Kx = mco x. Here, o> = Kjm is the angular fundamental frequency and m is the particle mass. In the case of dipole oscillations, m is the reduced dipole mass. What is the response of such an oscillator to an external electric field ... 

The susceptibility formula Eq. (2.20) can be used to describe the dispersions energy between macroscopic particles 1 and 2 along two alternative routes. We may replace each atom of particles 1 and 2 by a set of dipole oscillators and integrate the dispersion energy between any pair of atoms, or we may treat the particles as a whole as harmonic oscillators. 

Polarization of matter (1 ) can be describe as damped harmonic oscillator. When the electromagnetic field is sinusoidal the dipole oscillates around its position of equilibrium. This response is characterized by e(cp) ... 

By writing the frequencies of two dipole oscillators, and using the zero-point energy expression for a harmonic oscillator with the same proper frequency (equation 3.24), London [28] was able to derive the following expression for the dispersion energy between two molecules of polarizability a at a distance R ... 

The quantityis dimensionless and is the ratio of the strength of the transition to that of an electric dipole transition between two states of an electron oscillating in three dimensions in a simple harmonic way, and its maximum value is usually 1. 

Because T -> V energy transfer does not lead to complex formation and complexes are only formed by unoriented collisions, the Cl" + CH3C1 -4 Cl"—CH3C1 association rate constant calculated from the trajectories is less than that given by an ion-molecule capture model. This is shown in Table 8, where the trajectory association rate constant is compared with the predictions of various capture models.9 The microcanonical variational transition state theory (pCVTST) rate constants calculated for PES1, with the transitional modes treated as harmonic oscillators (ho) are nearly the same as the statistical adiabatic channel model (SACM),13 pCVTST,40 and trajectory capture14 rate constants based on the ion-di-pole/ion-induced dipole potential,... 

Physically, this formula describes the power dissipated by a harmonic oscillator (the emission dipole with moment t) as it is driven by the force felt at its own location from its own emitted and reflected electric field. PT is calculable given all the refractive indices and Fresnel coefficients of the layered model(12 33)... 

One may wonder whether a purely harmonic model is always realistic in biological systems, since strongly unharmonic motions are expected at room temperature in proteins [30,31,32] and in the solvent. Marcus has demonstrated that it is possible to go beyond the harmonic approximation for the nuclear motions if the temperature is high enough so that they can be treated classically. More specifically, he has examined the situation in which the motions coupled to the electron transfer process include quantum modes, as well as classical modes which describe the reorientations of the medium dipoles. Marcus has shown that the rate expression is then identical to that obtained when these reorientations are represented by harmonic oscillators in the high temperature limit, provided that AU° is replaced by the free energy variation AG [33]. In practice, tractable expressions can be derived only in special cases, and we will summarize below the formulae that are more commonly used in the applications. 

The general form of the energy of the harmonic oscillator indicates that the vibrational energy levels are equally spaced. Due to the vector character of the dipole transition operator, the transition between vibronic energy levels is allowed only if the following selection rule is satisfied ... 

Given the response of a single oscillator to a time-harmonic electric field, the optical constants appropriate to a collection of such oscillators readily follow. The induced dipole moment p of an oscillator is ex. If 91 is the number of oscillators per unit volume, the polarization P (dipole moment per unit... 

An additional point that should be considered is that in the harmonic oscillator approximation, the selection mle for transitions between vibrational states is Ay = 1, where v is the vibrational quantum number and Ay > 1, that is, overtone transitions, which involve a larger vibrational quantum number change, are forbidden in this approximation. However, in real molecules, this rule is slightly relaxed due to the effect of anharmonicity of the oscillator wavefunction (mechanical anharmonicity) and/or the nonlinearity of the dipole moment function (electrical anharmonicity) [55], affording excitation of vibrational states with Ay > 1. However, the absorption cross sections for overtone transitions are considerably smaller than for Ay = 1 transitions and sharply decrease with increasing change in Av. 

The derivation above may be generalized to wave functions other than electronic ones. By evaluation of transition dipole matrix elements for rigid-rotor and harmonic-oscillator rotational and vibrational wave functions, respectively, one arrives at the well-known selection rules in those systems that absorptions and emissions can only occur to adjacent levels, as previously noted in Chapter 9. Of course, simplifications in the derivations lead to many forbidden transitions being observable in the laboratory as weakly allowed, both in the electronic case and in the rotational and vibrational cases. 

The attenuation of die dipole of the repeat unit owing to thermal oscillations was modeled by treating the dipole moment as a simple harmonic oscillator tied to the motion of the repeat unit and characterized by the excitation of a single lattice mode, the mode, which describes the in-phase rotation of the repeat unit as a whole about the chain axis. This mode was shown to capture accurately the oscillatory dynamics of the net dipole moment itself, by comparison with short molecular dynamics simulations. The average amplitude is determined from the frequency of this single mode, which comes directly out of the CLD calculation ... 

In this paper we investigate the consequences of this addition to the original Montroll-Shuler equation. To keep it simple we retain the harmonic oscillator potential and the simple dipole-transition probabilities in the linear part of the equation and in the nonlinear part we restrict ourselves to the simple resonance transition... 

An alternative but not so general selection rule (it is restricted to the harmonic oscillator approximation) is that jVoI> / v5i dr is zero if dfiJdQ (evaluated for the equilibrium nuclear configuration) is zero, i.e. if there is no linear dependence of the dipole moment on the normal coordinate Q . 

What are the electric-dipole selection rules for a three-dimensional harmonic oscillator exposed to isotropic radiation ... 

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